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Structure of the set of solutions of an initial-boundary value problem for a parabolic partial differential equation in an unbounded domain. (English) Zbl 0862.35046

The author investigates, by means of an abstract evolution equation, the structure of the set of generalized solutions to a parabolic initial-boundary value problem in the classical space of continuous functions \(C([0,T]\times\overline\Omega)\). Precisely, let \(\Omega\subset\mathbb{R}^n\), \(n\geq2\), be a bounded domain with a smooth boundary \(\partial\Omega\). Let \[ A(x,D)v(x)= \sum_{|\alpha|\leq 2m}a_\alpha(x)D^\alpha v(x) \] be a strongly elliptic operator, and define the boundary operators \[ B_jv(x)= {\partial^{s+j}v(x)\over\partial\nu^{s+j}},\quad j=0,\dots,m-1 \] with \(0\leq s\leq m\) and \(\nu\) being the unit outward normal to \(\partial\Omega\). Consider the following problem \[ {\partial u(x,t)\over\partial t}+ A(x,D)u(x,t)= f(x,t,u,Du,\dots,D^{2m-1}u),\quad x\in\Omega,\quad t>0,\tag{\(*\)} \]
\[ B_ju(x,t)= 0,\quad x\in\partial\Omega,\quad t>0,\quad j=0,\dots,m-1,\quad u(x,0)=0,\quad x\in\Omega, \] where the function \(f(x,t,V)\) is measurable in \(x\), locally Hölder continuous in \(t\) and \(V\) with an exponent \(\theta\in(0,1]\) and such that for each \(T>0\) there exist positive functions \(\gamma_T,\beta_T\in L^p(\Omega)\) such that \(|f(x,t,V)|\leq\gamma_T(x)+ \beta_T(x)|V|\). The main result of the paper states that the set of all generalized solutions of the problem \((*)\) is an \({\mathcal R}_\delta\)-set in the space \(C([0,\infty)\times\overline\Omega)\). (A subset of a metric space \(X\) is an \({\mathcal R}_\delta\)-set iff it is homeomorphic to an intersection of a decreasing sequence of compact absolute retracts).

MSC:

35K35 Initial-boundary value problems for higher-order parabolic equations
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References:

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